a commissioned naval officer who ranks above a captain and
below a rear admiral; the lowest grade of admiral

From Webster's Revised Unabridged Dictionary (1913) [web1913]:

Commodore \Com"mo*dore`\, n. [Prob. a corruption of commander,
or Sp. comendador a knight of a military order who holds a
commandery; also a superior of a monastery, fr. LL.
commendare to command. Cf. {Commend}, {Command},
{Commander}.]
1. (U. S. Navy) An officer who ranks next above a captain;
sometimes, by courtesy, the senior captain of a squadron.
The rank of commodore corresponds with that of brigadier
general in the army.

2. (British Navy) A captain commanding a squadron, or a
division of a fleet, or having the temporary rank of rear
admiral.

3. A title given by courtesy to the senior captain of a line
of merchant vessels, and also to the chief officer of a
yachting or rowing club.

4. A familiar for the flagship, or for the principal vessel
of a squadron or fleet.